## 1. Introduction

Water in the ocean circulates around the globe on time scales of decades to millennia (e.g., Blanke et al. 2002; Johnson and Marshall 2004; Speich et al. 2007; Maltrud et al. 2010), and this global circulation has been related to climate variability on similar time scales (Broecker 1997; Knorr and Lohmann 2003). One of the branches of the global circulation for which the time scales have been studied is the “warm water return route” from the Indian Ocean to the North Atlantic Ocean (Speich et al. 2001). Weijer et al. (2002) have shown in a low-resolution (3.5° horizontal, 11 vertical layers) model that the salinity anomaly introduced in the thermocline and intermediate waters by Agulhas leakage into the South Atlantic is advected northward toward regions of deep convection, where it influences the strength of the Atlantic meridional overturning circulation (AMOC). Furthermore, Agulhas leakage appears to influence climate on paleoceanographic time scales (Peeters et al. 2004), presumably through its influence on the AMOC. An increasing trend in Agulhas leakage since the 1970s (Rouault et al. 2009; Biastoch et al. 2009) might thus suggest a strengthening of the AMOC after some time lag.

This time lag between Agulhas leakage and the AMOC is related, through changes in pressure gradients, to the radiation of Rossby and Kelvin waves in the Atlantic basin. Both Weijer et al. (2002) and van Sebille and van Leeuwen (2007) show that this radiative time scale is approximately 4 yr. However, Weijer et al. (2002) found a second (much slower) time scale in the system: the time scale with which salinity anomalies get advected through the Atlantic basin. Weijer et al. (2002) reported that the advective time scale from the South Atlantic to the regions of deep convection is between 25 and 50 yr. This advective time scale has not been verified by observations, because there are no Lagrangian multidecadal observational datasets available. Because the Lagrangian advection in the ocean is in part controlled by small-scale eddies (Griffa et al. 2007), which are not resolved in coarse-resolution models such as used by Weijer et al. (2002), results based on eddy-resolving models or (even better) solely on Lagrangian observations are valuable.

Here, we use both an observational and an eddy-resolving model dataset to study the advective time scales from the Agulhas region to the North Atlantic. The model data are from the ^{4} drifters have been released in the World Ocean, accounting for more than 19 million observations (Fig. 1), steadily increasing over the last 15 yr. Although global coverage is not uniform, 85% of the ocean surface has more than 100 observations per 1° grid cell.

The surface drifter dataset has proven immensely useful in studying the surface circulation of the ocean (e.g., Mariano et al. 2002; Lumpkin 2003), not least because the trajectories include all scales of motion. The dataset has been enhanced by creating synthetic trajectories based on multiple drifter trajectories (e.g., Özgökmen et al. 2000; Piterbarg 2001; Brambilla and Talley 2006), but most of these studies focus on regional spatial scales and relatively short time scales (a few years or less).

To study longer time scales than directly available from individual drifters, we introduce a technique to create “supertrajectories,” which are somewhat related to the extended trajectories of Brambilla and Talley (2006) but are more apt for studying the circulation on decadal to centennial time scales. In our technique, supertrajectories of any desired length can be formed from the global drifting buoy dataset (or other sets of short Lagrangian trajectories with sufficiently high spatial coverage) by essentially tying many short pieces of trajectories together, using a Monte Carlo approach.

The supertrajectories calculated from the surface drifter data are based solely on observations, without the uncertainties and assumptions inherent in numerical models. This advantage, however, comes at a price: the dataset only contains information on the two-dimensional (2D) surface circulation. The drifters are bound to the surface and cannot subduct and therefore do not perfectly follow water parcels. Instead, they converge in areas of mode water formation in the subtropics (Hanawa and Talley 2001) and diverge in the equatorial upwelling regions. Water flowing northward in the Atlantic Ocean goes through regions of net evaporation and net precipitation, both of which alter the density structure of the surface water. This has been quantified by Blanke et al. (2006), who found continuous water mass changes in the upper limb of the AMOC using numerical Lagrangian trajectories. A water parcel flowing from the Agulhas to the North Atlantic will thus likely not stay at the surface for the entire journey. Furthermore, because the surface drifters are in the Ekman layer, their motion is not only controlled by the large-scale pressure field but also to some extent by the Ekman transport and thereby by the local wind stress. The degree to which this 2D limitation on the surface drifter supertrajectories affects the advective pathways and time scales will be assessed by comparing them to results from the fully 3D supertrajectories computed in the high-resolution OFES.

## 2. Computation of supertrajectories

When the ocean is gridded into 1° grid cells, the number of available surface drifter trajectories is such that surface drifters cross the boundaries between two adjacent grid cells on average 9 times globally (Fig. 1c). These crossings can be used to calculate the probability of moving directly from a particular grid cell to any other. Each of these probabilities has an average time scale associated with it, computed as the difference in mean age in the grid cell before and after the crossing. Calculating these two quantities for each pair of grid cells, one obtains two (sparse) square matrices: one matrix that gives the probability of going from one particular grid cell directly to another and one matrix of the same shape that gives the average crossing time for this transition.

With these matrices, a large number (order 10^{4}) of trajectories starting at a particular grid cell can be computed using a Monte Carlo simulation. At each iteration of the growth of a trajectory, a next grid cell is randomly selected based on the probability matrix. The total time taken to complete the trajectories is then the sum of all crossing times. This approach is comparable to techniques used in statistical physics for numerically growing polymers to study their end-to-end length distribution (e.g., Newman and Barkema 1999). The statistics of these supertrajectories are robust, in the sense that the observed surface drifter trajectories and the supertrajectories are subsets of the same distribution.

The robustness of our technique can be demonstrated in a numerical model, where the number of drifters released is controlled (Figs. 2a,b). We have used surface horizontal velocity fields from a subregion of the 50-yr global

Assessment of the Monte Carlo supertrajectories technique in a numerical model. The probability that a grid cell is crossed for (a) the supertrajectories and (b) the directly integrated trajectories, both starting at the black dot and ending at the black line at 61°W. (c) The distribution of transit time from the black dot to 61°W obtained when directly integrating drifters (gray bars) is very similar to the distribution obtained from the Monte Carlo supertrajectories (lines). This is independent of the resolution on which the probability matrix is computed.

Citation: Journal of Physical Oceanography 41, 5; 10.1175/2011JPO4602.1

Assessment of the Monte Carlo supertrajectories technique in a numerical model. The probability that a grid cell is crossed for (a) the supertrajectories and (b) the directly integrated trajectories, both starting at the black dot and ending at the black line at 61°W. (c) The distribution of transit time from the black dot to 61°W obtained when directly integrating drifters (gray bars) is very similar to the distribution obtained from the Monte Carlo supertrajectories (lines). This is independent of the resolution on which the probability matrix is computed.

Citation: Journal of Physical Oceanography 41, 5; 10.1175/2011JPO4602.1

Assessment of the Monte Carlo supertrajectories technique in a numerical model. The probability that a grid cell is crossed for (a) the supertrajectories and (b) the directly integrated trajectories, both starting at the black dot and ending at the black line at 61°W. (c) The distribution of transit time from the black dot to 61°W obtained when directly integrating drifters (gray bars) is very similar to the distribution obtained from the Monte Carlo supertrajectories (lines). This is independent of the resolution on which the probability matrix is computed.

Citation: Journal of Physical Oceanography 41, 5; 10.1175/2011JPO4602.1

In the second experiment, using surface velocity fields from the same model, one numerical drifter was released in each grid cell and integrated forward in time for one year, the year 1980. This latter dataset was used to compute 10^{4} Monte Carlo supertrajectories originating from the same point that the individual drifters were released from in the first experiment. The directly integrated paths and the Monte Carlo supertrajectories are qualitatively similar (Figs. 2a,b), although the directly integrated trajectories form a somewhat smoother map. The trajectories in both datasets circulate predominantly in the Sargasso Sea, with very few trajectories crossing the Gulf Stream. This limited amount of mixing across the Gulf Stream was also observed in surface drifter data by Brambilla and Talley (2006) in their study of the mixing of surface water between the North Atlantic subtropical gyre and subpolar gyre. To compare transit times from the release point to other locations, we show in Fig. 2c the transit time distributions from the release point to the eastern edge of the domain (61°W) for both experiments. The distributions are remarkably similar for the two sets of trajectories, independent of the resolution on which the probability matrix is computed. This result confirms the validity of the supertrajectory technique.

A number of assumptions are made when computing Monte Carlo supertrajectories. One assumption is that all time scales of variability are captured in the Lagrangian dataset. This is obviously not the case with respect to the longer time scales, because the surface drifters have been measuring for less than 30 yr, whereas the model trajectories have been advected for one year. Both sets will thus fail to capture the complete cycle of, for instance, the Atlantic multidecadal oscillation (Enfield et al. 2001). However, this problem is not exclusive to the supertrajectories, because many Lagrangian-based estimates of circulation are limited by the time in which they are advected (i.e., the number of model years for which velocity data are available).

Another assumption that is made in the computation of Monte Carlo supertrajectories is that all drifters can be lumped together, regardless of season. In their computation of extended trajectories, Brambilla and Talley (2006) were careful to ensure that connected trajectories were in the same water mass to avoid seasonal biases. They did this by joining two surface drifter trajectories together only if their end points are spatially close enough together and the (deseasonalized) temperature difference between the two end points is sufficiently small. Ideally, here too the probability matrix should be computed based on the season or month in which the grid cell crossing takes place. However, this limits the amount of crossings available for calculating the probability matrix. For the application studied here, the effect of a seasonally varying probability matrix turns out to be of secondary importance (see section 3a).

## 3. Application to Agulhas leakage

### a. Observation-based 2D time scales

The surface drifter data are used to compute 10^{4} observational Monte Carlo supertrajectories between two sites in the Agulhas Current and the regions of deep convection in the North Atlantic Ocean (Fig. 3), ending either at the Greenland–Scotland Ridge or in the Labrador Sea. Many supertrajectories started in the Agulhas Current will take a very long time (on the order of millennia) to reach the North Atlantic, because they can circulate in the other oceans before entering the North Atlantic. Integrating these very long supertrajectories will take a lot of time, so, to cap computational time, supertrajectories that do not reach the North Atlantic within 100 yr are dismissed. This 100-yr cap is much longer than the advective time scale found by Weijer et al. (2002), and it turns out that the peak in supertrajectory transit time falls well before this cap.

Three examples of observational Monte Carlo supertrajectories starting in the Agulhas Current and ending at the Greenland–Scotland Ridge. (a) The shortest trajectory in the 10^{4} observational set, taking the direct route through the North Brazil Current, Caribbean Sea, and North Atlantic Current. (b) A trajectory with the most typical time scale of 35 yr, experiencing multiple circulations in the Indian and North Atlantic Ocean. (c) A trajectory with a much longer time scale, which circulates around Antarctica before crossing the equatorial Atlantic.

Citation: Journal of Physical Oceanography 41, 5; 10.1175/2011JPO4602.1

Three examples of observational Monte Carlo supertrajectories starting in the Agulhas Current and ending at the Greenland–Scotland Ridge. (a) The shortest trajectory in the 10^{4} observational set, taking the direct route through the North Brazil Current, Caribbean Sea, and North Atlantic Current. (b) A trajectory with the most typical time scale of 35 yr, experiencing multiple circulations in the Indian and North Atlantic Ocean. (c) A trajectory with a much longer time scale, which circulates around Antarctica before crossing the equatorial Atlantic.

Citation: Journal of Physical Oceanography 41, 5; 10.1175/2011JPO4602.1

Three examples of observational Monte Carlo supertrajectories starting in the Agulhas Current and ending at the Greenland–Scotland Ridge. (a) The shortest trajectory in the 10^{4} observational set, taking the direct route through the North Brazil Current, Caribbean Sea, and North Atlantic Current. (b) A trajectory with the most typical time scale of 35 yr, experiencing multiple circulations in the Indian and North Atlantic Ocean. (c) A trajectory with a much longer time scale, which circulates around Antarctica before crossing the equatorial Atlantic.

Citation: Journal of Physical Oceanography 41, 5; 10.1175/2011JPO4602.1

A probability density map of grid cell crossing for each supertrajectory reveals the typical surface path from the Agulhas Current to the North Atlantic (Fig. 4a). Most of the supertrajectories circulate in the Indian–Atlantic supergyre before crossing the equator in the North Brazil Current. The number of supertrajectories crossing the equatorial Atlantic outside of the North Brazil Current is negligible. Most of the supertrajectories then follow the Loop Current into the Gulf Stream. The probability of circulating in the subtropical gyre in the North Atlantic is large, and when a supertrajectory gets into the North Atlantic Current it travels toward the Greenland–Scotland Ridge. The part of the journey in the North Atlantic is in good agreement with the results of the extended trajectories calculated by Brambilla and Talley (2006). In a reproduction of one of their experiments using the observational supertrajectories, only 6% of the supertrajectories that start in the Gulf Stream (at 40°N, 72°W) reach 55°N in the subpolar gyre within 600 days, a result remarkably similar to the 5% of the extended trajectories computed by Brambilla and Talley (2006) that reach the subpolar gyre from the Gulf Stream region within the same amount of time.

Surface advection time scales from the observational supertrajectories. (a) Map of the chance that a supertrajectory that starts in the Agulhas Current (circles) and ends in one of the areas of deep-water formation (black lines in the North Atlantic) crosses a particular grid cell. The ocean grid cells without trajectories are shaded white. (b) Distribution of the transit time of the 2D observational Monte Carlo supertrajectories of the surface circulation. The bulk of the observational supertrajectories reach the North Atlantic in 25–40 yr, whereas the fastest supertrajectory reaches the North Atlantic in 4 yr.

Citation: Journal of Physical Oceanography 41, 5; 10.1175/2011JPO4602.1

Surface advection time scales from the observational supertrajectories. (a) Map of the chance that a supertrajectory that starts in the Agulhas Current (circles) and ends in one of the areas of deep-water formation (black lines in the North Atlantic) crosses a particular grid cell. The ocean grid cells without trajectories are shaded white. (b) Distribution of the transit time of the 2D observational Monte Carlo supertrajectories of the surface circulation. The bulk of the observational supertrajectories reach the North Atlantic in 25–40 yr, whereas the fastest supertrajectory reaches the North Atlantic in 4 yr.

Citation: Journal of Physical Oceanography 41, 5; 10.1175/2011JPO4602.1

Surface advection time scales from the observational supertrajectories. (a) Map of the chance that a supertrajectory that starts in the Agulhas Current (circles) and ends in one of the areas of deep-water formation (black lines in the North Atlantic) crosses a particular grid cell. The ocean grid cells without trajectories are shaded white. (b) Distribution of the transit time of the 2D observational Monte Carlo supertrajectories of the surface circulation. The bulk of the observational supertrajectories reach the North Atlantic in 25–40 yr, whereas the fastest supertrajectory reaches the North Atlantic in 4 yr.

Citation: Journal of Physical Oceanography 41, 5; 10.1175/2011JPO4602.1

The distribution of transit times from the Agulhas to the North Atlantic of these observational supertrajectories is broad and shows a peak at 30–40 yr (Fig. 4b). This time scale is roughly similar to the advective time scale found by Weijer et al. (2002) in a low-resolution global model. The shortest transit time of any of the 10^{4} supertrajectories is 4 yr, which is similar to the radiation time scale associated with the group velocity of Rossby waves crossing the Atlantic, as reported by Weijer et al. (2002) and van Sebille and van Leeuwen (2007). This agreement between the two time scales is probably a coincidence, because it is unclear why the radiation time scale should set the minimum advection time scale.

In computing the probability matrix, all surface drifters exiting a grid box are treated equally. In regions with a strong seasonal cycle in the surface circulation, such as the equatorial Atlantic, this might bias the results. One way to account for the seasonal variability in the circulation is to create a time-dependent probability matrix. By using a probability matrix that includes only the drifter crossings in the season corresponding to the age of the supertrajectory, a seasonal cycle can be included. The disadvantage of a time-dependent probability matrix is that the number of crossings is reduced. In the case of the surface drifters, working with four different probability matrices for the four seasons decreases the average number of crossings between two grid cells to only two. Supertrajectories can be computed using such a probability matrix, but their accuracy is much lower because some paths are not possible any more, an artifact created by the low sampling. This effect will increase with the length of the path, because the cumulative chance of a path becoming stuck in a grid cell increases with pathlength. Nevertheless, a comparison of the transit time distribution of observational supertrajectories with and without a seasonally varying probability matrix does not reveal large differences (Fig. 5). The shorter transit times are somewhat more likely and the longer time scales are somewhat less likely when the seasons are included, and the average transit time decreases by 4 yr. However, both the peak in the distribution at 30 yr and the minimum transit time of 4 yr are similar to the nonseasonal supertrajectory distribution. The amount of paths performing a circumpolar loop is smaller for the seasonal probability matrix, but the pathways in the Atlantic Ocean are nearly identical (not shown).

Comparison of transit times for the observational Monte Carlo supertrajectories using the original time-invariant probability matrix (black line) and when the probability matrix is computed from the crossings that occur within a particular season (gray line). Transit times are somewhat shorter when seasonal effects are taken into account, but both distributions peak around 30 yr and the difference in the cumulative distribution is always less than 10 yr.

Citation: Journal of Physical Oceanography 41, 5; 10.1175/2011JPO4602.1

Comparison of transit times for the observational Monte Carlo supertrajectories using the original time-invariant probability matrix (black line) and when the probability matrix is computed from the crossings that occur within a particular season (gray line). Transit times are somewhat shorter when seasonal effects are taken into account, but both distributions peak around 30 yr and the difference in the cumulative distribution is always less than 10 yr.

Citation: Journal of Physical Oceanography 41, 5; 10.1175/2011JPO4602.1

Comparison of transit times for the observational Monte Carlo supertrajectories using the original time-invariant probability matrix (black line) and when the probability matrix is computed from the crossings that occur within a particular season (gray line). Transit times are somewhat shorter when seasonal effects are taken into account, but both distributions peak around 30 yr and the difference in the cumulative distribution is always less than 10 yr.

Citation: Journal of Physical Oceanography 41, 5; 10.1175/2011JPO4602.1

To find 10^{4} supertrajectories that reached either of the two sites in the North Atlantic within 100 yr, 10^{6} trajectories had to be started from the Agulhas Current. This is because most trajectories do not reach the North Atlantic within 100 yr. The surface drifters (the basis of the probability matrix) have the tendency to accumulate in the subtropical gyres, where surface water converges and subducts. Evidence of this influence of subduction zones on the supertrajectories can be seen in the extensive recirculations in the subtropical gyres in Fig. 4a. It might thus be that the advective time scales following water masses are shorter than those obtained from the surface circulation alone.

### b. Model-based 3D time scales

To assess to what extent the subduction zones affect the results from the observational supertrajectories, a second experiment is conducted using three-dimensional numerical supertrajectories. Note that it is not possible to study the advective time scales up to 100 yr from Lagrangian floats in OFES directly, because the model run is only integrated for 50 yr. Numerical floats are integrated using the full-depth velocity field, including vertical velocity, from one year of global high-resolution OFES output (the same model configuration as used in section 2). The floats are released uniformly over the global ocean on a 50-m vertical by 0.2° horizontal grid. These three-dimensional trajectories are then used to create a full-depth three-dimensional probability matrix of the global circulation. Within this probability matrix, a similar set of 10^{4} supertrajectories is calculated that start in the Agulhas Current at 15-m depth and reach either of the two end sections at any depth within 100 yr.

The set of numerical supertrajectories is expected to better represent Lagrangian circulation than the observational supertrajectories, because of its three-dimensional character. The numerical supertrajectories can be viewed as more accurately following water mass parcels, which can subduct within the thermocline and be advected at subsurface depths. The supertrajectories based on the 3D circulation in the OFES have a slightly larger probability of reaching the North Atlantic in 100 yr (2%, compared to 1% for the observational supertrajectories).

The pathways of the numerical supertrajectories are rather different from the surface observational supertrajectories (cf. Fig. 6a with Fig. 4a) and agree better with the Lagrangian pathways of the upper limb of the AMOC found by Blanke et al. (2006) in an eddy-permitting model. As anticipated, because the supertrajectories are able to subduct, the pathways in the Atlantic subtropical gyres are less diffuse in this numerical set. In the Southern Hemisphere, the probability map shows an extension of the Indian–Atlantic supergyre to the Pacific Ocean and also some supertrajectories looping the Antarctic Circumpolar Current and entering the Atlantic Ocean via the “cold water route” through Drake Passage (Rintoul 1991; Speich et al. 2001).

Three-dimensional advection time scales from the numerical supertrajectories (cf. Fig. 4 of the observational supertrajectories). (a) Map of the chance that a supertrajectory that starts in the Agulhas Current (circle) and ends in one of the areas of deep-water formation (black lines in the North Atlantic) crosses a particular grid cell. The ocean grid cells without trajectories are shaded white. (b) Distribution of the transit time of the 3D numerical Monte Carlo supertrajectories. The bulk of the supertrajectories reach the North Atlantic in 10–25 yr, shorter than the surface-tied observational supertrajectories. The fastest supertrajectory reaches the North Atlantic in 4 yr, similar to that time scale in the observations.

Citation: Journal of Physical Oceanography 41, 5; 10.1175/2011JPO4602.1

Three-dimensional advection time scales from the numerical supertrajectories (cf. Fig. 4 of the observational supertrajectories). (a) Map of the chance that a supertrajectory that starts in the Agulhas Current (circle) and ends in one of the areas of deep-water formation (black lines in the North Atlantic) crosses a particular grid cell. The ocean grid cells without trajectories are shaded white. (b) Distribution of the transit time of the 3D numerical Monte Carlo supertrajectories. The bulk of the supertrajectories reach the North Atlantic in 10–25 yr, shorter than the surface-tied observational supertrajectories. The fastest supertrajectory reaches the North Atlantic in 4 yr, similar to that time scale in the observations.

Citation: Journal of Physical Oceanography 41, 5; 10.1175/2011JPO4602.1

Three-dimensional advection time scales from the numerical supertrajectories (cf. Fig. 4 of the observational supertrajectories). (a) Map of the chance that a supertrajectory that starts in the Agulhas Current (circle) and ends in one of the areas of deep-water formation (black lines in the North Atlantic) crosses a particular grid cell. The ocean grid cells without trajectories are shaded white. (b) Distribution of the transit time of the 3D numerical Monte Carlo supertrajectories. The bulk of the supertrajectories reach the North Atlantic in 10–25 yr, shorter than the surface-tied observational supertrajectories. The fastest supertrajectory reaches the North Atlantic in 4 yr, similar to that time scale in the observations.

Citation: Journal of Physical Oceanography 41, 5; 10.1175/2011JPO4602.1

The circulation pattern in the equatorial Atlantic is more intricate in the model than in the drifter observations, with a much increased probability of interior pathways. These result primarily from eastward advection into the interior from the western boundary in the thermocline layers (particularly in the Equatorial Undercurrent) and subsequent upwelling and meridional transport in the surface layer. The circulation pattern in the subpolar North Atlantic is also considerably different in the 3D model calculation. Although 90% of the surface observational supertrajectories end at the Greenland–Scotland Ridge, that value is only 5% for the numerical supertrajectories. Inspection of the circulation pattern suggests that this is due to the numerical supertrajectories becoming entrained into the westward-flowing Greenland Current just before reaching the Greenland–Scotland Ridge. This happens predominantly to the subsurface supertrajectories, because the supertrajectories that end in the Labrador Sea are on average more than 3 times deeper in the North Atlantic than the ones that end on the Greenland–Scotland Ridge.

The peak of the transit time distribution is 10–25 yr, considerably shorter in the numerical supertrajectories than in the observational ones (cf. Fig. 6b with Fig. 4b). The transit times shorter than 30 yr are more probable in the numerical set, and consequently the longer time scales are less probable. This difference is related to the numerical supertrajectories more easily crossing the subtropical gyres by subducting. The fastest trajectory, however, has a similar transit time as in the observational set, at 4 yr. The 10–25-yr peak in model transit time is also shorter than the results from the low-resolution model study of Weijer et al. (2002). This might be related to the generally higher velocities in eddying models. Both these eddies and the better-resolved boundary currents result in a higher probability of relatively high velocities compared to low-resolution models, and it is therefore possible to take a faster route through the ocean basins.

The 100-yr cap in supertrajectory integration length suffices to compute the peak transit time in both datasets but is too short to compute the mean transit time. Nevertheless, the distributions for time scales longer than the peak transit time appear to approximately follow an exponential decay (Figs. 4b, 6b). If the transit time distributions are extended with an exponential fit, the mean transit times are 81 ± 6 yr for the observational set of supertrajectories and 54 ± 5 yr for the numerical set. Similar to the peak in transit time, the mean transit time is longer in the 2D data than in the 3D data.

## 4. Discussion and outlook

We have studied the time it takes surface waters to get advected from the Agulhas Current in the Indian Ocean to either the Labrador Sea or the Greenland–Scotland ridge. For this purpose, a technique is introduced to compute trajectories of any desired length from short Lagrangian trajectories such as those from observational surface drifting buoys. It is shown that these supertrajectories have similar statistical properties as the Lagrangian trajectories themselves. With these supertrajectories, investigation of the basin-scale circulation on centennial time scales is possible using either observational (surface drifter) data or numerical model data. The technique is particularly suited for applications when only Lagrangian data are available, such as the case with surface drifter trajectories.

The time scales and pathways of supertrajectories based on the full three-dimensional velocity in a numerical model are 10 yr shorter than supertrajectories from surface drifter data and less persistent in the subtropical gyres, an indication of the influence of convergence and subduction on the global circulation. However, the fastest time scale is at 4 yr, equal in the 2D and 3D sets of supertrajectories.

Recently, there have been a number of studies debating how similar the paths and time scales obtained from Lagrangian methods are to those obtained from tracers. Bower et al. (2009), Zhang (2010), and Lozier (2010) found that the circulation of North Atlantic Deep Water in the subtropical North Atlantic Ocean is different in a Lagrangian context than in an Eulerian one, whereas Brambilla and Talley (2006) showed that far fewer surface drifting buoys cross the North Atlantic subpolar front than expected from Eulerian considerations on the upper limb of the AMOC. On the other hand, a study by van Sebille et al. (2010) on the Eulerian and Lagrangian assessment of the circulation in the Agulhas region suggests that these two frameworks can lead to consistent estimates of Agulhas leakage. It is not the goal of this study to provide definitive answers on the dichotomy between the global circulation in Eulerian and Lagrangian frameworks, but there are some results from this study that can aid in the discussion. The transit time scales found with the Monte Carlo supertrajectories are shorter than the Eulerian estimates by Weijer et al. (2002), but that might be related to the model resolution. On the other hand, the dominant pathway through the Atlantic (via the southern subtropical gyre, the Brazil Current, the Loop Current, the northern subtropical gyre, and finally the North Atlantic Current) agrees with streamfunction maps computed from for instance altimetry (e.g., Blanke et al. 2006; Maximenko et al. 2009).

We can foresee many more applications of the Monte Carlo supertrajectory technique. One of these applications is the study of intergyre mixing on a global scale, similar to the study by Brambilla and Talley (2006) on surface mixing between the subtropical and subpolar Atlantic gyres (see also section 3a). In other applications, subsurface Argo floats can be used instead of the surface drifters to populate the probability matrix and study circulation at middepth, provided that the bias in displacement during ascend and descend can be accounted for (Willis and Fu 2008). Furthermore, model trajectories can be compared with observational supertrajectories to assess model skill (van Sebille et al. 2009). But perhaps the most tantalizing application is the use of supertrajectories from high-resolution numerical models to obtain a Lagrangian view of the global three-dimensional ocean circulation on millennial time scales, as was done by, for instance, Drijfhout et al. (1996), using a low-resolution model.

## Acknowledgments

This research was supported by the U.S. National Science Foundation under Award OCE0241438. The OFES simulation was conducted on the Earth Simulator under the support of JAMSTEC. This manuscript benefited greatly from the comments of several reviewers.

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